The sign-dependent Iwasawa main conjecture for weight interlacing strings

Let $[?]$ be the Hecke algebra of a $p$-adic family of Galois representations $[?]$. For a tuple $[?]$ of interlacing strings, let $[?]$ be the associated Selmer complex and let $[?]$ be the corresponding $p$-adic measure.

Sign-dependent Iwasawa main conjecture. If $[?]$, then $[?]$ is torsion and

$$\operatorname{char}_{\mathbb{I}}\widetilde{\mathrm{H}}^2_\square(\mathcal{K},\boldsymbol{T})=(\mathcal{L}_p^\square)^2.$$

If $[?]$, then $[?]$ has rank $1$ over $[?]$, and there is $[?]$ such that

$$\operatorname{char}_{\mathbb{I}}\widetilde{\mathrm{H}}^2_\square(\mathcal{K},\boldsymbol{T})_{\mathrm{tors}}=\operatorname{char}_{\mathbb{I}}\left(\frac{\widetilde{\mathrm{H}}^1_\square(\mathcal{K},\boldsymbol{T})}{\mathbb{I}\cdot\boldsymbol{z}}\right)^2.$$

The two cases reflect the root number: sign $+1$ predicts nonvanishing central values and a torsion Selmer group, whereas sign $-1$ predicts central vanishing and a rank-one Selmer group. The framework is proposed in the source and remains open.